The Slant height is 12 in.
The base edge is 20 in.
What is the surface area of the roof of the doll house?
what's the shape?
To find the surface area of the roof of the dollhouse, we need to calculate the area of each triangle face and add them together. The roof of a dollhouse is in the shape of a triangular pyramid.
Given:
Slant height = 12 in
Base edge = 20 in
Step 1: Find the height of the pyramid using the Pythagorean theorem.
The height (h) of the pyramid is the perpendicular distance from the apex to the base.
Using the Pythagorean theorem:
h = √(slant height^2 - base edge^2/4)
h = √(12^2 - 20^2/4)
h = √(144 - 100)
h = √44
h ≈ 6.63 in
Step 2: Calculate the area of each triangular face.
The area of a triangle can be calculated using the formula: Area = (base × height) / 2.
Each triangular face has a base edge of 20 in and a height of 6.63 in.
Area of one face = (20 × 6.63) / 2
Area of one face ≈ 66.3 in²
Step 3: Calculate the total surface area of the roof.
Since the roof of the dollhouse is a pyramid with four triangular faces, we need to multiply the area of one face by 4.
Total surface area = 4 × 66.3 in²
Total surface area ≈ 265.2 in²
Therefore, the surface area of the roof of the dollhouse is approximately 265.2 square inches.
To find the surface area of the roof of the doll house, we need to use the formula for the surface area of a pyramid. The surface area of a pyramid is calculated by adding the area of the base to the sum of the areas of the triangular faces.
Let's break it down step by step:
Step 1: Find the area of the base.
The base edge is given as 20 in, and since it is a square base, all sides have the same length. The formula for the area of a square is side length squared, so the area of the base is 20 in × 20 in = 400 square inches.
Step 2: Find the area of the triangular faces.
The slant height of the pyramid is given as 12 in. The triangular faces are isosceles triangles with two equal sides (the slant height) and one base edge (the base edge of the square base).
To find the area of each triangular face, we need to calculate the height of the triangle. Let's use the Pythagorean theorem to find the height:
Using the Pythagorean theorem, we have:
height^2 + (1/2 * base edge)^2 = slant height^2
Let's plug in the numbers:
height^2 + (1/2 * 20 in)^2 = 12 in^2
height^2 + 10^2 = 12^2
height^2 + 100 = 144
height^2 = 144 - 100
height^2 = 44
height = sqrt(44)
height ≈ 6.63 in
Now that we have the height, we can calculate the area of each triangular face:
Area of a triangle = (base × height) / 2
Area of each triangular face = (20 in × 6.63 in) / 2
Area of each triangular face ≈ 66.3 square inches
Step 3: Calculate the total surface area.
Since there are four triangular faces on a pyramid, we can multiply the area of one triangular face by four.
Total surface area = 4 × 66.3 square inches
Total surface area ≈ 265.2 square inches
Therefore, the surface area of the roof of the doll house is approximately 265.2 square inches.
If it's a square pyramid, then each face is a triangle with area
1/2 * 20 * 12